Twice differentiability of solutions to fully nonlinear parabolic equations near the boundary

TL;DR

This paper establishes boundary regularity results for viscosity solutions of non-convex fully nonlinear parabolic equations, extending known elliptic regularity results to the parabolic setting.

Contribution

It proves $ ext{H}^{2+ ext{alpha}}$ regularity near the boundary for solutions to non-convex fully nonlinear parabolic equations, a significant extension of elliptic regularity theory.

Findings

Boundary regularity of solutions is established.

Extension of elliptic regularity results to parabolic equations.

Viscosity solutions exhibit $ ext{H}^{2+ ext{alpha}}$ regularity near the boundary.

Abstract

In this paper, we prove $\mathcal{H}^{2+\alpha}$ regularity for viscosity solutions to non-convex fully nonlinear parabolic equations near the boundary. This constitutes the parabolic counterpart of a similar $C^{2, \alpha}$ regularity result due to Silvestre and Sirakov proved in [15] for solutions to non-convex fully nonlinear elliptic equations.